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<p>This article proposes a family of iterative methods without memory, including two-step fourth order, three-step sixth order, and <i>m</i>-step 2<i>m</i>-order iterative methods. Through error analysis, we rigorously proved the convergence order of these methods. These iterative methods without memory only require computing the inverse of a first-order divided-difference operator, thus having lower computational costs. On this basis, with the help of the first-order divided-difference operator, we designed eight acceleration parameters to extend the iterative method without memory to a based iterative method with memory. Specifically, the highest convergence order of the two-step iterative method with memory can reach 4.302, the highest convergence order of the three-step iterative method with memory can reach 7.274, and the highest convergence order of the <i>m</i>-step iterative method with memory can reach <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((2m+1+\sqrt{4m^{2}+60m-127})/2 \)</EquationSource> </InlineEquation>. We further calculated the computational cost of these iterative methods and compared their computational efficiency with known methods. In the numerical experiment, this paper used the proposed iterative method to solve typical nonlinear systems, including Hammerstein type integral equations, boundary value problems, and common nonlinear system problems. The experimental results not only verified the correctness of the theoretical conclusions, but also demonstrated significant advantages in both computational efficiency and computation time of the proposed method.</p>

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High-efficiency multi-step derivative-free iterative methods with and without memory for solving nonlinear systems

  • 期刊论文

  • 发表日期：

    2026年3月14日

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  • Xiaofeng Wang,
  • Jia Yu

摘要

This article proposes a family of iterative methods without memory, including two-step fourth order, three-step sixth order, and m-step 2m-order iterative methods. Through error analysis, we rigorously proved the convergence order of these methods. These iterative methods without memory only require computing the inverse of a first-order divided-difference operator, thus having lower computational costs. On this basis, with the help of the first-order divided-difference operator, we designed eight acceleration parameters to extend the iterative method without memory to a based iterative method with memory. Specifically, the highest convergence order of the two-step iterative method with memory can reach 4.302, the highest convergence order of the three-step iterative method with memory can reach 7.274, and the highest convergence order of the m-step iterative method with memory can reach \((2m+1+\sqrt{4m^{2}+60m-127})/2 \) . We further calculated the computational cost of these iterative methods and compared their computational efficiency with known methods. In the numerical experiment, this paper used the proposed iterative method to solve typical nonlinear systems, including Hammerstein type integral equations, boundary value problems, and common nonlinear system problems. The experimental results not only verified the correctness of the theoretical conclusions, but also demonstrated significant advantages in both computational efficiency and computation time of the proposed method.